Unformatted text preview: spectrum Of a field K. Thanks to (2.3.4.1.11), the homomorphism (2.3.4.3.1) has source and target of the same dimension, as does (2.3.4.3.2), and both homomorphisms have the same kernel, namely F i+1(3 fcnon i. Q.E.D. (2.3.4.4) Corollary. Assumptions as in (2.3.2), and n>O fixed, suppose that the map h(i) figuring in (2.3.4.2.4) is an isomorphism. Then: (2.3.4.4.1) The Csmodule u+a~F~ox~t(R"f,(f2}/s(logD))) is locally free, and its formation commutes with arbitrary change of base S'~ S. (2.3.4.4.2) The canonical mapping F~on i 1 RnJ, (f2~c/s (log D)) , FZo ~ ' @ F i + 1 ~ FZo ~ i 1 is an isomorphism. Proof. It suffices to prove (2.3.4.4.2), since by hypothesis F~o ~ ~~/F~o: i is locally free, and its formation commutes with arbitrary change of base S'~ S. The composite mapping, formed from the bottom half of the diagram (2.3.4.2.4), F:o~ '1 (R" f, (t2~:/s (log D))) hi, +x), R"f, (~]:s (log D))/F '+ 2 F~o~ '(R"J, (Q'x/s (log D))) ~hc0 ~ R.f, (f2~,/s (log D))/F '+1...
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 Fall '11
 NormanKatz
 Equivalence relation, Isomorphism, Homomorphism, Congruence relation, Algebraic Solutions of Differential Equations

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